She fears with every youngster under 18 included in the scheme, overstretched staff could fail to spot signs of abuse, leading to another case like Baby Peter or Victoria Climbie.
Ms Davidson spoke out as she revealed the Scottish Conservatives are to launch a fresh bid to stop the measure, which is included in the Children and Young People (Scotland) Act already passed by Holyrood.
Children’s Minister Aileen Campbell said that the Scottish Government had “no intention” of halting the legislation, which assigns a ‘’named person’’ – such as a teacher or health visitor – to look out for the welfare of children under 18.
A legal challenge against the policy at Scotland’s top civil court failed earlier this year but the No To Named Persons campaign group has secured a hearing at the Supreme Court in London in March.
Writing in a Sunday newspaper, Ms Davidson said: “I think it is wrong that the government of a country can impose a guardian to watch over your child without consent and without reason to believe that child needs protection.
“It is a grave violation of the right to a family life, a provision which includes the right to the holding of private information.”
“By stretching resources so thinly that officials are desperately trying to cover every child in the country, it takes time, attention and resources away from those children who need it most. She added: “And this is my real concern. What happens when something goes wrong?
“When a child somewhere in Scotland who desperately needs help and intervention is overlooked as their ‘named person’ has dozens of other perfectly healthy and happy children on the books and can’t take the time for the child who needs it?
“It is not beyond imagining we could see a case of neglect or abuse in Scotland – a Victoria Climbie or Baby Peter – which results in tragedy.”
Authorities in the London borough of Haringey were criticised after the death of 17-month-old Peter Connelly in 2007, after he suffered more than 50 injuries over an eight month period when he was being seen by children’s services.
The Scottish Government stressed while the Named Person service would be available to all children and their parents, they would not be legally obliged to use it or take up any help offered.